Discussion Papers No. 235, October 1998 Statistics Norway, Research Department
Erik Biørn, Kjersti-Gro Lindquist and Terje Skjerpen
Random Coefficients and Unbalanced Panels:
An Application on Data from Norwegian Chemical Plants
Abstract:
A framework for analyzing substitution and scale properties, and technical change from plant-level panel data is presented. Focus is on comparing the constant and random coefficient specification of
the substitution and scale parameters and investigating the potential variation of the parameters across firms. Characteristics of the model framework are (i) an equation system consisting of a three-factor translog cost function and the corresponding cost-share equations, (ii) random firm specific heterogeneity in coefficients, and (iii) a Maximum Likelihood procedure allowing for unbalanced panel data. The empirical results, based on data from Norwegian chemical plants, indicate substantial firm specific heterogeneity in substitution and scale properties.
Keywords: Panel Data. Random Coefficients. Unbalanced Data. Heterogeneity. Production technology
JEL classification: C 33, D21, D24, L65
Acknowledgement: Paper presented at the Eighth International Conference on Panel Data, Göteborg, June 1998, and at the Econometric Society European Meeting, Berlin, August - September 1998. We are grateful to Petter Frenger, Jan Larsson, and conference participants for comments.
Address: Erik Biørn, University of Oslo, Department of Economics, P.O. Box 1095 Blindern, N- 0317 Oslo, Norway. E-mail: [email protected]
Kjersti-Gro Lindquist, Statistics Norway, Research Department.
E-mail: [email protected]
Terje Skjerpen, Statistics Norway, Research Department. E-mail: [email protected]
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1 Introduction
A common challenge in microeconometric analyses of economic relationships is how to treat heterogeneity concerning the form of the relationships across the micro units. Such heterogeneity may be modelled and analyzed when panel data are available, unlike the situation when only cross section data are at hand. However, even in a panel data context, most researchers have tended to assume a common coecient structure, possibly allowing for unit specic (or time specic) dierences in intercept terms of the equations (`xed' or `random' eects) only. If the heterogeneity has a more complex form, this modelling approach may lead to inecient (and even inconsistent) estimation of the slope coecients, and hence invalid inference.
A more general approach is to allow for heterogeneity also in slope coecients of the equation. The challenge then becomes how to obtain a model which is suciently exible while avoiding overparametrization. The xed eects slope coecient approach, in which each unit has its distinct coecient vector, with no a priori assumptions made about its variation between units, is very exible, but may easily suer from this degrees of freedom problem. The random coecients approach, in which specic assumptions are made about the distribution from which the unit specic coecients are drawn, is far more parsimonious in this respect. At the same time, the expectation vector in this distribution represents, in a precise way, the coecients of an average unit, while its second order moments matrix gives easily interpretable measures of the degree of heterogeneity. The random coecients approach may also be considered a particular way of representing disturbance heteroskedasticity in panel data analysis, since the random eects enter multiplicatively to the covariates of the equation.
While there is a growing body of methodological articles in the econometric literature dealing explicitly with this random coecient problem for balanced panel data situations, far less has been done with unbalanced panel data. This is somewhat surprising, since in practice the latter is rather the rule than the exception. Because working with complete panels is mathematically more convenient, a common procedure for empirical researchers is to leave out the units for which the time series are incomplete and use the balanced sub-sample of the original, less tidy data-set. This may, however, involve a of loss of eciency, see Matyas and Lovrics (1991) and Baltagi and Chang (1994).
In this paper, we consider a general framework for analyzing the production process, i.e., factor substitution, scale properties and technical change, from unbalanced plant- level panel data. The paper discusses the importance of choosing a random versus the more common constant coecients specication of plant heterogeneity in econometric analyses of factor demand systems. The translog cost function approach suggested by
Christensen et al. (1971, 1973) is acommodated to an application on plant-level panel data for Norwegian chemical industries for the years 1972 { 1993.
The model framework of the present analysis has the following basic characteristics:
(i) A translog cost-function with three variable inputs and technical change and derived cost-share equations. Capital input is treated as a fourth, `quasi-xed' input, reecting the assumed long lead time needed to build new capacity in the sector under consid- eration, and this mechanism cannot be adequately represented by static optimization.
(ii) Plant specic heterogeneity in some coecients of the cost function is allowed for and treated as random. (iii) The model is designed for an unbalanced plant-level panel data set. While (i) is rather standard, the combination of (ii) and (iii) is not, at least in applied econometrics. Mixed regression models with unbalanced design, however { both uni- and multivariate { have, to some extent, been discussed in the statistical literature
denoted as
X
andK
respectively, and both, like the three input prices, are treated as exogenous variables, i.e., capital is treated as a `quasi-xed' factor. A deterministic trend, , representing, inter alia, technical change is included.The production technology, represented by its dual cost function, is assumed to be a translog function in (
X;K;;Q
L;Q
E;Q
M) [see, e.g., Christensen, Jorgenson, and Lau (1971, 1973) and Jorgenson (1986)]:ln
C
= 0+X lnX
+K lnK
++PgglnQ
g + PgXg(lnX
)(lnQ
g) +PgKg(lnK
)(lnQ
g) + Pgg lnQ
g+ 12PgPhgh(lnQ
g)(lnQ
h) + 12XX(lnX
)2+12KK(lnK
)2+122+
XK(lnX
)(lnK
) +X lnX
+K lnK; g;h
=L;E;M:
Cost minimization with respect to
V
g, conditional onQ
g (g
=L;E;M
),X
andK
gives, according to Shephard's lemma (@C=@Q
g =V
g), the following expression for the optimal cost share of factorg
:Q
gV
gC
=@
lnC
@
lnQ
g =g+XglnX
+KglnK
+PhghlnQ
h+g:
The homogeneity and symmetry conditions on the cost function and the adding-up of cost shares imply
Pg
g = 1;
PgXg=PgKg=Pgg =Pggh= 0;
gh=hg:
We represent these restrictions in the model by letting M = 1,L,E;
XM =,XL,XE;
KM =,KL,KE;
M =,L,E;
LM =,LL,LE;
EM =,EE ,LE;
MM =LL+ 2LE+EE;
and can write the cost function and the cost-share equations of labour and energy asln(
C=Q
M) = 0+XlnX
+KlnK
++Pggln(Q
g=Q
M) (1)+ Pg
Xg(lnX
)(ln(Q
g=Q
M)) +PgKg(lnK
)(ln(Q
g=Q
M)) + Pggln(Q
g=Q
M) +12PgPhgh(ln(Q
g=Q
M))(ln(Q
h=Q
M)) + 12XX(lnX
)2+12KK(lnK
)2+122+
XK(lnX
)(lnK
) +XlnX
+KlnK; g;h
=L;E;
Q
gV
gC
=g+XglnX
+KglnK
+g +Phghln(Q
h=Q
M); g;h
=L;E;
(2)
The cost elasticity of output, capital, and the rate of increase of cost with time are, respectively,
(
@
lnC
)=
(@
lnX
) ="
X =X+XXlnX
+PgXgln(Q
g=Q
M) +XKlnK
+X;
(3)(
@
lnC
)=
(@
lnK
) ="
K =K+KKlnK
+PgKgln(Q
g=Q
M) +XKlnX
+K;
(4) (@
lnC
)=
(@
) ="
= + +Pggln(Q
g=Q
M) +KlnK
+XlnX:
(5)
Cross-price and own-price elasticities of substitution in the demand for factor
g
with respect to the price of factorh
, dened as the Slutsky analogues (output constrained price elasticities of input quantities), are"
gh=8
>
>
>
<
>
>
>
:
ghs
g +s
h; g
6=h;
ggs
g +s
g,1; g
=h;
(6)
which satisfy Ph
"
gh = 0. The corresponding (symmetric) Allen-Uzawa elasticities of substitution are gh=hg ="
ghs
h =8
>
>
>
<
>
>
>
:
ghs
gs
h + 1; g
6=h;
ggs
2g + 1,s
1g; g
=h:
(7)
Denoting the cost shares as
s
g = (Q
gV
g)=C
(g
=L;E
), and using lower case letters to symbolize logarithms, i.e.,c
= lnC;x
= lnX;q
g = lnQ
g etc., the cost-share equations of labour and energy, (2), can be written as1s
L=L+XLx
+KLk
+L+LL(q
L,q
M) +LE(q
E,q
M) +u
L;
(8)s
E =E+XEx
+KEk
+E +LE(q
L,q
M) +EE(q
E,q
M) +u
E;
(9)and the cost function, (1), as
(
c
,q
M) = 0+Xx
+Kk
+ +L(q
L,q
M) +E(q
E,q
M) (10)+
XLx
(q
L,q
M) +XEx
(q
E,q
M) +KLk
(q
L,q
M) +KEk
(q
E,q
M) +L(q
L,q
M) +E(q
E,q
M)+ 12
LL(q
L,q
M)2+LE(q
L,q
M)(q
E ,q
M) +12EE(q
E,q
M)2 + 12XXx
2+12KKk
2+122+XKxk
+Xx
+Kk
+u
C;
1To these equations may be added, for symmetry reasons, the corresponding equation for materials, but to avoid singularity of the disturbance covariance matrix, it is omitted from the econometric model.
Using, as here, a Maximum Likelihood estimation procedure, this involves no loss of eciency.
where we have added the disturbances
u
L,u
E, andu
C.The data are from an unbalanced panel, in which the plants are observed in at least 1 and at most
P
years. We assume that the plants are arranged in groups according to the number of years the plants are observed. This will be convenient when presenting the estimation procedure (cf. Section 3). LetN
p be the number of plants which are observed in exactlyp
years (not necessarily the same and not necessarily consecutive), let (ip
) index thei
'th plant among those observed inp
years (i
= 1;:::;N
p;p
= 1;:::;P
), and lett
index the observation number (t
= 1;:::;p
). The total number of plants in the panel isN
=PPp=1N
pand the total number of observations isn
=PPp=1N
pp
. To capture heterogeneity of the technology, some coecients are allowed to be plant dependent and treated as random.Two model classes will be considered:
Mo del A:The two cost-share equations, (8) and (9).
Mo del B:The two cost-share equations and the cost function, (10).
Identication of the scale properties of the technology and the trend eects is possible within the full Model B only. The substitution properties can be identied from both models. Model A contains 11 and Model B contains 21 (xed or random) coecients.
Since the latter incorporates more prior information, it leads to more ecient parameter estimation within a full Maximum Likelihood procedure, provided that all restrictions are valid.
The two model versions can be written compactly as
y
(ip)t=X(ip)t(ip)+u(ip)t
; p
= 1;:::;P
;i
= 1;:::;N
p;t
= 1;:::;p;
(11)
where(ip) is the coecient vector of plant (
ip
), in which at least some elements are be random and the other are xed constants common to all plants. Model A is characterized byX 0=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
1 0
0 1
x
00
x
k
00
k
(
q
L,q
M) 0 (q
E,q
M) (q
L,q
M)0 (
q
E ,q
M) 00
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
=2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
L E XL XE KL KE LL LE EE L E3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
0= [
s
Ls
E];
0= [u
Lu
E];
and Model B by
X 0=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 0 1
0 0
x
0 0
k
0 0
1 0 (
q
L,q
M)0 1 (
q
E,q
M)x
0x
(q
L,q
M)0
x x
(q
E,q
M)k
0k
(q
L,q
M)0
k k
(q
E ,q
M)(
q
L,q
M) 0 12(q
L,q
M)2 (q
E,q
M) (q
L,q
M) (q
L,q
M)(q
E ,q
M)0 (
q
E ,q
M) 12(q
E,q
M)2 0 (q
L,q
M)0
(
q
E ,q
M)0 0 12
x
20 0 12
k
20 0 12
20 0
xk
0 0
x
0 0
k
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
=2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 X K L E XL XE KL KE LL LE EE L E XX KK XK X K3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
;
y
0= [
s
Ls
Ec
,q
M];
u0= [u
Lu
Eu
C];
In the applications, the following model versions will be considered:Mo del A1: All 11 coecients are xed.
Mo del A2: The coecients
L;
E are random. The other 9 coecients are xed.Mo del B1: All 21 coecients are xed.
Mo del B2: The coecients
0;
L;
E are random. The other 18 coecients are xed.Mo delB3: The coecients
0;
X;
K;
;
L;
E are random. The other 15 coecients, representing second-order terms in the cost function, are xed.In addition, a `constant returns to scale' version of Model B2, denoted as B2R is also considered. Model B3 implies that in the two cost-share equations, only the intercepts are random, whereas the cost equation from which they are derived, have ve random slope coecients, and a random intercept. Hence, both the cost elasticity function with respect to output and the substitution elasticity function, conditional on the exogenous variables, will contain random coecients.
The model is formally a system of
G
regression equations with random coecients and with a total ofK
(xed or random) coecients. In Model A,G
= 2;K
= 11, inModel B,
G
= 3;K
= 21. The (G
1) vector of observations of the regressands in theG
equations from plant (ip
), observationt
isy(ip)t, and the corresponding (G
K
) regressor matrix isX(ip)t. The (K
1) coecient vector of plant (ip
) is
(ip)=+(ip)
;
(12)where is the common expectation vector of (ip) for all plants, and(ip) a zero mean vector specic to plant (
ip
). By inserting (11) in (12), theG
equations for plant (ip
), observationt
, can thus be written asy
(ip)t=X(ip)t+(ip)t
;
(ip)t=X(ip)t(ip)t+u(ip)t:
(13)We further assume that
X
(ip)t
;
u(ip)t;
(ip) are all stochastically independent;
(14)and that
u
(ip)tI IN(0G1
;
u);
(ip)I IN(0K1;
);
(15)whereI INsignies independently, identically, normally distributed,0m;n is a (
m
n
) zero matrix and
u=
2
6
6
4
11u 1uG... ...
uG1 uGG3
7
7
5
;
=2
6
6
4
11
1K
... ...
K1 KK
3
7
7
5
:
The latter two matrices may be singular, reecting for instance that some of the coe- cients may be xed. In all model versions we consider below,u is a full positive denite (
G
G
) matrix, while the (K
K
) matrix has reduced rank and is partioned as=
2
4
0
0 0 3
5
;
where in, e.g., Model A2,u =
2
4
uLL uLE uEL uEE3
5
;
=2
4
LL LE EL EE3
5
;
and in, e.g., Model B3,u=
2
6
6
4
uLL uLE uLC uEL uEE uEC uCL uCE uCC3
7
7
5
;
=2
6
6
6
6
6
6
6
6
6
6
6
4
00 0X 0K 0 0L 0E X0 XX XK X XL XE K0 KX KK K KL KE 0 X K L E L0 LX LK L LL LE E EX EK E EL EE3
7
7
7
7
7
7
7
7
7
7
7
5
:
We stack the
p
realizations from plant (ip
) iny
(ip)=
2
6
6
4 y
(ip)1
...
y
(ip)p
3
7
7
5
;
X(ip)=2
6
6
4 X
(ip)1
...
X
(ip)p
3
7
7
5
;
u(ip)=2
6
6
4 u
(ip)1
...
u
(ip)p
3
7
7
5
;
(ip)=2
6
6
4
(ip)1
...
(ip)p
3
7
7
5
;
which, in general, have dimensions (Gp
1), (Gp
K
), (Gp
1), and (Gp
1), respectively.Then we can write (13) as
y
(ip)=X(ip)+(ip)
;
(ip)=X(ip)(ip)+u(ip):
(16)It follows from (14), (15), and (16) that
All(ip)jX(ip)are stochastically independent, and (ip)jX(ip)N(0Gp;1
;
(ip));
(17)where(ip) is the (
Gp
Gp
) matrix
(ip) = X(ip)X(0ip)+Ipu
:
(18)We see from (18) that the `gross disturbance' vector (ip) exhibits a particular kind of heteroskedasticity.
3 Estimation procedure and data
The joint log-density function of plant (
ip
), i.e. ofy(ip)conditional on X(ip), isL
(ip)=,Gp
2 ln(2
), 12 lnj(ip)j, 1
2[y(ip),X(ip)]0,1(ip)[y(ip),X(ip)]
;
so that by utilizing the ordering of the observations in theP
groups, we can write the log-likelihood function of all observations ony conditional on all observations onX asL
=XPp=1 Np
X
i=1
L
(ip)=,Gn
2 ln(2
), 1 2P
X
p=1 Np
X
i=1lnj(ip)j (19)
,
12
P
X
p=1 Np
X
i=1[y(ip),X(ip)]0,1(ip)[y(ip),X(ip)]
:
The Maximum Likelihood (ML) estimators of (;
u;
) are obtained by maximizingL
with respect to (the unknown elements of) these parameter matrices, as given in Section 2.2 The structure of this problem is more complicated than the ML problem for systems2The soluton conditions may be simplied by concentrating L over and maximizing the resulting function with respect to the unknown elements of thematrices.
of regression equations in more standard situations with balanced panel data sets and xed slope coecients for two (related) reasons. First, the variousy,X, andmatrices have dierent number of rows, reecting the dierent number of observations of the plants in the panel. Although the dimensions ofu, and in (18) are the same for all plants, the dimensions ofX(ip), and hence of(ip), dier. Second, dierent plants have dierent disturbance covariance matrices, since (ip) depends on X(ip) when is non-zero, i.e., when at least one of the coecients (in addition to the intercepts) are random.
Primarily we use data from the Manufacturing Statistics database of Statistics Nor- way, supplemented, to a minor extent by data from the Norwegian National Accounts.
All industries classied under SIC-code 351 Manufacture of Industrial Chemicals are in- cluded. The data set is unbalanced and covers
T
= 22 years, with a total number of plantsN
= PpN
p = 88, and a total number of observationsn
= PpN
pp
= 1101, so that on average, the plants are observed in 11 { 12 years. Of these plants,N
22 = 30 are observed in all the 22 years { representing 660 observations or about 60 percent of the data set { andN
1 = 16 are observed in one year only. All times series used for the estimation and testing reported below are contiguous, i.e., plants for which there are gaps in the time series in the original database, 220 observations in all, are excluded. The output measure is tonnes output. This implies that systematic, gradual quality changes in output over time will be represented by the trend variable in the model. The capital input data are constructed from information on re insurance values, in combination with information on gross investment ows from the Manufacturing Statistics and in the National Accounts. Details on the data and data construction are given in the Appendix.4 Empirical results
Maximum Likelihood estimates of the six models described in Section 2 have been ob- tained by using the PROC MIXED-procedure in SAS/STAT software [SAS (1992)].
In addition to the estimated elements of the vector, cf. equation (12), Tables 1 and 3 include the coecients calculated by using the adding-up conditions described in Section 2. Apart from the results for (the restrictive) Model B2R, the tables show that several coecient estimates are relatively stable across models. This supports our stochastic assumptions, which implies that neglecting coecient heterogeneity does not cause inconsistent estimates of asymptotically. However, there is a loss of eciency.
A clear majority of the coecient estimates are signicant according to the asymptotic standard-error estimates.
Tables 2 and 4 give some measures of overall model t. Both the Akaike and the
Schwartz Bayesian information criteria support the most exible models, i.e., A2 among the A-models and B3 among the B-models. Furthermore, it is evident that substantial explanatory power is lost when the output-elasticity a priori is set to one, which is the case for Model B2R. The Log-Likelihood values of the dierent models are also given in Tables 2 and 4, and the Log-Likelihood Ratios of interest can be calculated from the tables. These test-statistics are, however, not asymptotically
2-distributed under the null, because the coecients then are on the border of the admissible parameter space, see Shin (1995, p. 321). Thus, for making formal inference, other test procedures are needed.For a discussion of statistical tests for mixed linear models see Khuri et al. (1998).
To facilitate the economic interpretation of the models, we have calculated various elasticities. In Table 5, own-price and cross-price elasticities of (variable) input demand are reported. These are the analogues of the Slutsky-elasticities from consumer demand, and the cross-price elasticities are not symmetric. In addition, we also report the Allen- Uzawa elasticities, which are symmetric, see Table 6. By construction, the two sets of elasticities have the same sign. In accordance with the chosen translog functional form, the elasticities will be functions not only of the estimated coecients, but also of the exogenous variables. Typically, the elasticities will therefore vary across plants and over time. In Tables 5 and 6, the dierent elasticities are reported at the overall sample means of the exogenous variables. The elasticities in Models A2, B2, B3 and B2R are functions of both the exogenous variables and the random coecients, and these models will in general predict variation in elasticities even across plants with the same exogenous input. When calculating the elasticities reported in the tables, we use the estimated expectations of the random coecients. Because the predicted (variable) cost shares depend on unknown coecients, all the estimated elasticities in Tables 5 and 6 are non-linear functions of the coecients. To calculate standard deviations, a rst order Taylor-expansion of the non-linear functions is utilized (cf. Kmenta (1986)).
According to the results in Table 5, none of the variable inputs are price-elastic. This is in line with the ndings in Lindquist (1995), where a dynamic translog factor-demand system assuming xed eects is estimated on Norwegian manufacturing plant-level panel data. Except for energy in Model B1, the own-price elasticities in the six models are all less than 1 in absolute value. From Table 6 it is seen that all the cross-price Allen- Uzawa elasticities are positive (at the global mean of the exogenous variables), which means that all the three variable inputs in average are substitutes. The Allen-Uzawa own-price elasticity for energy is fairly high in absolute value in all models and higher than the estimates usually reported in studies of input demand. The price elasticities are relatively equal in the A- and B-models. Even the price elasticties from Model B2R
are rather similiar to those obtained for the other models. Hence, in our case, little gain is obtained in adding the cost function to the cost-share equations if the sole interest is in factor-price elasticities. However, in this paper we are also interested in the scale properties of the technology, which can only be obtained within a framework where the cost function is an integral part of the model, as in our B-models.
The rst row of Table 7 contains the output elasticity in the B-models. This elasticity, like the price elasticities, depends on the value of the exogenous variables (except in Model B2R where it is restricted to 1 a priori), and again we use the overall mean of the exogenous variables and the estimated expected values of the random coecients as a reference point when calculating the elasticities. The output elasticity is between 0.25 and 0.3 in the random coecient models B2 and B3. In Model B1, in which all coecients are assumed to be plant-invariant, the estimated output elasticity is about 0.15. The inverse of the output elasticity can be interpreted as `a variable-input scale elasticity' of the underlying (three factor) production function. These estimated `economies of scale eects' are rather high compared with the unitary elasticity assumed in Model B2R. The loss of explanatory power for Model B2R is closely linked to higher output elasticity for this model than for the other B-models.
In the second row of Table 7, we report the elasticity of variable costs with respect to an increase in the capital stock calculated at the global empirical mean. To be well behaved, the cost function should be non-increasing and convex in the levels of any xed factor, cf. Brown and Christensen (1981, pp. 217 { 218). The intuition is that as capital stock increases, one needs less variable inputs to maintain the given level of output. This will accordingly reduce variable costs. However, for Models B1, B2 and B3 this elasticity has wrong sign, implying that an increase in the capital stock increases variable costs at a given level of output. The capital elasticity has the correct sign in Model B2R, however, although it is not signicantly dierent from zero according to the asymptotic t-value.
In the last row of Table 7, we report the derivative of the logarithm of variable costs with respect to the trend variable. The trend variable is included in order to pick up the eect of technological progress, which may be both neutral and non-neutral. This derivative should be of negative sign, since technological progress means that a given output level can be obtained with less variable inputs. As for the response to an increase in the capital stock, we get a positive sample mean estimate of this derivative for Models B1, B2 and B3, whereas the correct negative sign is obtained for Model B2R, and the eect is signicant.
In order to study the robustness of the results obtained for Model B3 with respect to the production technology, several alternative calculations have been performed. First,
insignicant coecients have been constrained to zero and the model reestimated follow- ing a general to specic modelling approach. Second, an alternative capital measure was constructed, based on re insurance values, while the rst measure was very much based on data on investments. However, the results using this alternative capital measure came out very similiar to the results in Table 7, and they are therefore not reported.
Our interpretation of the `theory-inconsistent' results is that we are unable to properly identify simultaneously the eect of changes in production, capital stock, and technolog- ical progress on variable costs. This may again be due to our proxies for these two variables. In our context there are no generally accepted way to calculate the capital stock. A lot of dierent capital stock measures can be established, and empirically, there are no operational way to distinguish between them. Our results are, however, robust to which of the alternative measures we include in our information set (see Appendix).
Technological change is proxied by a deterministic time trend which is, admittedly, a very rough way of picking up technological progress, and the need for additional information about technological progress is highly desirable. The problem of identifying the dier- ent properties of the production process is not particular to our study. Morrison (1988, p. 278) for instance, reports within a Generalized Leontief framework that \empirical researchers have found it dicult to identify independently the impacts of technology, quasi-xed inputs and returns to scale".
In Table 8, we report the estimated covariance matrix of the random coecient-vector for Models A2, B2, B3, and B2R, respectively. These matrices are of dierent dimensions, because the number of random coecients varies across the models. The covariance matrix of the genuine disturbances of the six estimated models are given in Table 9. The covariance matrices of the genuine error-terms are of dimension (22) and (33) for the A- and B-models, respectively. All the estimated covariance matrices satisfy the positive- deniteness requirement, since all the calculated eigenvalues of the dierent matrices are positive. Both the A- and B-models which do not allow for coecient heterogeneity have higher estimated variances of the genuine disturbances than the more exible models.
This is as expected when coecient heterogeneity is important, because Models A1 and B1 are then misspecied and the estimated variances of the disturbances are inuenced by the coecient heterogeneity. By comparing the estimated covariance matrices for Model B2R with those of Model B2, we see that constraining the output elasticity to 1 has a signicant impact on both the covariance matrix for the random coecients and the covariance matrix of the genuine error-terms. However, the submatrix consisting of only the second order moments of the genuine errors in the two cost-share equations looks very similiar to that in Model B2.
Table 1. Coefficient estimates and standard deviations in A-models
Model A11 Model A22
Coefficient Value Std. dev. Value Std. dev.
γL 0.3351 0.0465 0.2261 0.0557
γE 0.0825 0.0290 -0.0027 0.0354
γM 0.5824 0.0622 0.7766 0.0711
βXL -0.0257 0.0024 -0.0171 0.0022
βXE 0.0115 0.0019 0.0060 0.0016
βXM 0.0142 0.0034 0.0111 0.0027
βKL 0.0021 0.0029 0.0030 0.0049
βKE 0.0018 0.0023 0.0069 0.0033
βKM -0.0040 0.0042 -0.0099 0.0064
γLL 0.0445 0.0088 0.0466 0.0057
γLE -0.0192 0.0045 -0.0089 0.0026
γLM -0.0253 0.0103 -0.0377 0.0062
γEE -0.0112 0.0044 0.0053 0.0024
γEM 0.0303 0.0068 0.0036 0.0035
γMM -0.0050 0.0146 0.0341 0.0080
βτL 0.0003 0.0007 0.0008 0.0004
βτE 0.0017 0.0006 0.0015 0.0003
βτM -0.0020 0.0011 -0.0023 0.0005
1 Model A1: None of the coefficients are assumed to be random.
2 Model A2: γL , γE and γM are the expectations of γL(i,p) , γE(i,p) and γM(i,p) respectively.
Table 2. Overall measures of fit in A-models
Model A1 Model A2
Number of estimated parameters 14 17
Log-likelihood value 1546.398 2877.179
Akaike’s information criterion 1543.398 2871.179
Schwartz’s Bayesian criterion 1534.853 2854.088
